I would like to find the asymptotic behavior of the integral
$$\int_0^1 (1-t^2)^{-1/2} e^{-nt} \,dt$$
for large $n$. It seems reasonably obvious that the integral goes to zero. At least it is bounded; the integral is between $0$ and
$$\int_0^1 (1-t^2)^{-1/2} \,dt = \pi/2.$$
I am just learning asymptotic methods and I'm having trouble even approaching this. I thought that Laplace's method might be appropriate but only the case of $\int_{-\infty}^{\infty}$ is discussed in the books I have.
Full, detailed steps would be greatly appreciated. My goal is to try to estimate a slightly more complicated integral.